Sharp comparison and maximum principles via horizontal normal mapping in the Heisenberg group
Analysis of PDEs
2016-02-15 v3
Abstract
In this paper we solve a problem raised by Guti\'errez and Montanari about comparison principles for convex functions on subdomains of Heisenberg groups. Our approach is based on the notion of the sub-Riemannian horizontal normal mapping and uses degree theory for set-valued maps. The statement of the comparison principle combined with a Harnack inequality is applied to prove the Aleksandrov-type maximum principle, describing the correct boundary behavior of continuous convex functions vanishing at the boundary of horizontally bounded subdomains of Heisenberg groups. This result answers a question by Garofalo and Tournier. The sharpness of our results are illustrated by examples.
Keywords
Cite
@article{arxiv.1305.5638,
title = {Sharp comparison and maximum principles via horizontal normal mapping in the Heisenberg group},
author = {Zoltán M. Balogh and Andrea Calogero and Alexandru Kristály},
journal= {arXiv preprint arXiv:1305.5638},
year = {2016}
}
Comments
Published in: J. Funct. Anal. 269 (2015), no. 9, 2669-2708